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Asset Pricing With No Exogenous Probability Measure

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  • Gianluca Cassese

Abstract

In this paper, we propose a model of financial markets in which agents have limited ability to trade and no probability is given from the outset. In the absence of arbitrage opportunities, assets are priced according to a probability measure that lacks countable additivity. Despite finite additivity, we obtain an explicit representation of the expected value with respect to the pricing measure, based on some new results on finitely additive measures. From this representation we derive an exact decomposition of the risk premium as the sum of the correlation of returns with the market price of risk and an additional term, the purely finitely additive premium, related to the jumps of the return process. We also discuss the implications of the absence of free lunches.

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  • Gianluca Cassese, 2008. "Asset Pricing With No Exogenous Probability Measure," Mathematical Finance, Wiley Blackwell, vol. 18(1), pages 23-54, January.
    • Handle: RePEc:bla:mathfi:v:18:y:2008:i:1:p:23-54
    DOI: 10.1111/j.1467-9965.2007.00321.x
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    1. Itzhak Gilboa & David Schmeidler, 1995. "Case-Based Decision Theory," The Quarterly Journal of Economics, Oxford University Press, vol. 110(3), pages 605-639.
    2. J. Michael Harrison & Stanley R. Pliska, 1981. "Martingales and Stochastic Integrals in the Theory of Continous Trading," Discussion Papers 454, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    3. Kreps, David M., 1981. "Arbitrage and equilibrium in economies with infinitely many commodities," Journal of Mathematical Economics, Elsevier, vol. 8(1), pages 15-35, March.
    4. Harrison, J. Michael & Pliska, Stanley R., 1983. "A stochastic calculus model of continuous trading: Complete markets," Stochastic Processes and their Applications, Elsevier, vol. 15(3), pages 313-316, August.
    5. Jarrow, Robert A & Rosenfeld, Eric R, 1984. "Jump Risks and the Intertemporal Capital Asset Pricing Model," The Journal of Business, University of Chicago Press, vol. 57(3), pages 337-351, July.
    6. Markowitz, Harry M & Perold, Andre F, 1981. "Portfolio Analysis with Factors and Scenarios," Journal of Finance, American Finance Association, vol. 36(4), pages 871-877, September.
    7. Kent Daniel & David Hirshleifer & Avanidhar Subrahmanyam, 1998. "Investor Psychology and Security Market Under- and Overreactions," Journal of Finance, American Finance Association, vol. 53(6), pages 1839-1885, December.
    8. Schweizer, Martin, 1992. "Martingale densities for general asset prices," Journal of Mathematical Economics, Elsevier, vol. 21(4), pages 363-378.
    9. Merton, Robert C, 1973. "An Intertemporal Capital Asset Pricing Model," Econometrica, Econometric Society, vol. 41(5), pages 867-887, September.
    10. Clark, Stephen A., 2000. "Arbitrage approximation theory," Journal of Mathematical Economics, Elsevier, vol. 33(2), pages 167-181, March.
    11. Kent D. Daniel & David Hirshleifer & Avanidhar Subrahmanyam, 2001. "Overconfidence, Arbitrage, and Equilibrium Asset Pricing," Journal of Finance, American Finance Association, vol. 56(3), pages 921-965, June.
    12. Harrison, J. Michael & Kreps, David M., 1979. "Martingales and arbitrage in multiperiod securities markets," Journal of Economic Theory, Elsevier, vol. 20(3), pages 381-408, June.
    13. Harrison, J. Michael & Pliska, Stanley R., 1981. "Martingales and stochastic integrals in the theory of continuous trading," Stochastic Processes and their Applications, Elsevier, vol. 11(3), pages 215-260, August.
    14. Clark, Stephen A., 1993. "The valuation problem in arbitrage price theory," Journal of Mathematical Economics, Elsevier, vol. 22(5), pages 463-478.
    15. Detemple, Jerome B & Zapatero, Fernando, 1991. "Asset Prices in an Exchange Economy with Habit Formation," Econometrica, Econometric Society, vol. 59(6), pages 1633-1657, November.
    16. Schmeidler, David, 1989. "Subjective Probability and Expected Utility without Additivity," Econometrica, Econometric Society, vol. 57(3), pages 571-587, May.
    17. Alexandre Ziegler, 2000. "Optimal Portfolio Choice under Heterogeneous Beliefs," Review of Finance, European Finance Association, vol. 4(1), pages 1-19.
    18. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    19. K. J. Arrow, 1964. "The Role of Securities in the Optimal Allocation of Risk-bearing," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 31(2), pages 91-96.
    20. Rubinstein, Mark, 1994. "Implied Binomial Trees," Journal of Finance, American Finance Association, vol. 49(3), pages 771-818, July.
    21. Mark Rubinstein., 1994. "Implied Binomial Trees," Research Program in Finance Working Papers RPF-232, University of California at Berkeley.
    22. Duffie, Darrell & Epstein, Larry G, 1992. "Asset Pricing with Stochastic Differential Utility," Review of Financial Studies, Society for Financial Studies, vol. 5(3), pages 411-436.
    23. Gianluca Cassese, 2005. "A Note On Asset Bubbles In Continuous-Time," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 8(04), pages 523-536.
    24. Back, Kerry, 1991. "Asset pricing for general processes," Journal of Mathematical Economics, Elsevier, vol. 20(4), pages 371-395.
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    Citations

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    Cited by:

    1. Gianluca Cassese, 2021. "Complete and competitive financial markets in a complex world," Finance and Stochastics, Springer, vol. 25(4), pages 659-688, October.
    2. Alexander Cox & Jan Obłój, 2011. "Robust pricing and hedging of double no-touch options," Finance and Stochastics, Springer, vol. 15(3), pages 573-605, September.
    3. Gianluca Cassese, 2014. "Option Pricing in an Imperfect World," Papers 1406.0412, arXiv.org, revised Sep 2016.
    4. Zhaoxu Hou & Jan Obloj, 2015. "On robust pricing-hedging duality in continuous time," Papers 1503.02822, arXiv.org, revised Jul 2015.
    5. Travis Fisher & Sergio Pulido & Johannes Ruf, 2015. "Financial Models with Defaultable Num\'eraires," Papers 1511.04314, arXiv.org, revised Oct 2017.
    6. Matteo Burzoni & Marco Frittelli & Marco Maggis, 2016. "Universal arbitrage aggregator in discrete-time markets under uncertainty," Finance and Stochastics, Springer, vol. 20(1), pages 1-50, January.
    7. Vladimir Vovk, 2012. "Continuous-time trading and the emergence of probability," Finance and Stochastics, Springer, vol. 16(4), pages 561-609, October.
    8. Matteo Burzoni & Marco Frittelli & Marco Maggis, 2016. "Universal arbitrage aggregator in discrete-time markets under uncertainty," Finance and Stochastics, Springer, vol. 20(1), pages 1-50, January.
    9. Zhaoxu Hou & Jan Obłój, 2018. "Robust pricing–hedging dualities in continuous time," Finance and Stochastics, Springer, vol. 22(3), pages 511-567, July.
    10. Travis Fisher & Sergio Pulido & Johannes Ruf, 2017. "Financial Models with Defaultable Numéraires," Working Papers hal-01240736, HAL.
    11. Lorenzo Bastianello & Alain Chateauneuf & Bernard Cornet, 2022. "Put-Call Parities, absence of arbitrage opportunities and non-linear pricing rules," Papers 2203.16292, arXiv.org.
    12. Gianluca Cassese, 2017. "Asset pricing in an imperfect world," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 64(3), pages 539-570, October.
    13. Jan Obłój & Johannes Wiesel, 2021. "A unified framework for robust modelling of financial markets in discrete time," Finance and Stochastics, Springer, vol. 25(3), pages 427-468, July.
    14. Christian Bender & Sebastian Ferrando & Alfredo Gonzalez, 2021. "Model-Free Finance and Non-Lattice Integration," Papers 2105.10623, arXiv.org.
    15. Gianluca Cassese, 2008. "Finitely Additive Supermartingales," Journal of Theoretical Probability, Springer, vol. 21(3), pages 586-603, September.

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